Okay, I’ve put this one off for long enough…
A while back, I explained why social welfare functions don’t exist. In this post, I also described how neoclassical economics has abandoned the idea of cardinal welfare functions in favor of ordinal utility functions due to the fact that cardinal utility functions cannot withstand legitimate criticisms.
As a quick reminder, a utility function is a numerical mapping of preferences. I would apply a value of 100 for a good milkshake and 5 for a bad milkshake, for example. An ordinal utility function has it that the good milkshake is preferred to the bad milkshake, but this is all we can derive from the numbers provided; we cannot tell how much I would prefer a good milkshake to a bad one. On the other hand, a cardinal utility function actually allows us to form ratios and calculate this: I would gain 100 units of utility from a good milkshake, but only 5 from a bad milkshake.
Now, the problem here is that there is no measure of utility as such, so we cannot create cardinal utilities, leaving us with only ordinal utilities.
But that is not the end of the story. Instead of using some made-up ‘util’ measure, some economists attempt to create a cardinal utility function by using money as a proxy for utility. You see this often in mixed-strategy game theory, as a matter of fact, but also in other contexts. This is what I would call a pseudo-cardinal utility function: you assume that utilities as such are still ordinal, but you use a cardinal proxy to determine the changes in utility. An easy way to think of it is like this: you currently have a utility of x. If I give you a $5 transfer, you will have a utility of x + y, where y is the utility gained from an additional $5. If I give you a $10 transfer, you will have a utility of x + 2y.
But if you look closely, there’s an inherent problem here: money is subject to diminishing returns. Imagine it this way: if you tax a rich man to the tune of $1000, it bothers him less than if you tax a man whose net value is $1000 by the same amount. Thus, the utility derived from an additional $5 is actually dependent upon the current utility, and you need to measure the utility in a consistent manner over values. In other words, you’re back to a cardinal utility function.
So basically, pseudo-cardinal utility functions either avoid the fact that there are diminishing returns to money or they implicitly assume a cardinal utility function to piggyback off of. In either case, they are not acceptable for serious use, any more than calculating the number of utils is acceptable.